Dic₅₃₂₅ #36319017

order	21300
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic	Dic₅₃₂₅
unambiguous semidirect product	C₃⋊Dic₁₇₇₅
unambiguous semidirect product	C₇₁⋊Dic₇₅

Splittings

derived, Hall	C₃×C₂₅×C₇₁ ⋊ C₄
Hall	C₂₅×C₇₁ ⋊ Dic₃
Hall	C₃×C₇₁ ⋊ Dic₂₅
Hall	C₃×C₂₅ ⋊ Dic₇₁
Sylow	C₇₁ ⋊ Dic₇₅
Sylow	C₂₅ ⋊ Dic₂₁₃
Sylow	C₃ ⋊ Dic₁₇₇₅

Subgroups

	subgroup	G/H
center, hypercenter	C₂	D₅₃₂₅
commutator, hypocenter	C₃×C₂₅×C₇₁	C₄
Frattini	C₂×C₅	D₁₀₆₅
Fitting, cosocle	C₂×C₃×C₂₅×C₇₁	C₂
socle, prime radical	C₂×C₃×C₅×C₇₁	D₅
LCS	G ▷ C₃×C₂₅×C₇₁
UCS	C₁ ◁ C₂

36 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 103
transitive degree	μ_t(G) = 21300

A minimal faithful permutation representation — 2 generators on 103 points:

(1,2)(3,4)(5,6,7)(8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32)(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103);(1,3,2,4)(6,7)(9,32)(10,31)(11,30)(12,29)(13,28)(14,27)(15,26)(16,25)(17,24)(18,23)(19,22)(20,21)(34,103)(35,102)(36,101)(37,100)(38,99)(39,98)(40,97)(41,96)(42,95)(43,94)(44,93)(45,92)(46,91)(47,90)(48,89)(49,88)(50,87)(51,86)(52,85)(53,84)(54,83)(55,82)(56,81)(57,80)(58,79)(59,78)(60,77)(61,76)(62,75)(63,74)(64,73)(65,72)(66,71)(67,70)(68,69)

Polycyclic presentation:

2,3,5,71,2,5|a^2=e,c^5=f^4,b^a=b^2,c^a=c^4f,d^a=d^70,f^a=f^4

Conjugacy classes & character table

5328 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₁₀₆₅₀, Dic₁₅₉₇₅
maximal normal subgroup of	Dic₁₀₆₅₀
maximal quotient of	Dic₁₅₉₇₅

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(21300, 9);

Dic5325 #36319017